Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,14,20,16,22,15,21)(11,23,26,33,28,38,27,37)(12,29,13,32,31,19,30,18)(17,24,34,25,36,40,35,39);(1,3)(2,5)(4,8)(6,7)(9,11,16,28)(10,17,22,36)(12,25,31,39)(13,24,30,40)(14,27,15,26)(18,23,32,38)(19,33,29,37)(20,35,21,34);(3,8)(5,7)(9,12,15,30,16,31,14,13)(10,18,20,29,22,32,21,19)(11,24,27,39,28,40,26,25)(17,33,34,38,36,37,35,23)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=g,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130208, #130209, #130248, … |
| maximal normal subgroup of | #130208, #130209, #130248, … |
| maximal quotient of | #122735, #122740, #122742, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4880);